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In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively.

The Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula.

There are two kinds of these polynomials. The first kind $T_n$ is defined by the recurrence $$\begin{align} T_0(x)&=1\\ T_1(x)&=x\\ T_{n+1}(x)&=2xT_n(x)-T_{n-1}(x) \end{align}$$ The second kind $U_n$ is defined by the same recurrence, but with $U_1(x)=2x$.

These polynomials also satisfy the trigonometric identities $$T_n(\cos\theta)=\cos(n\theta)\qquad U_n(\cos\theta)\sin\theta=\sin(n+1)\theta.$$

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